We now consider that our active particle moves through a (one-dimensional) dilute gas of light particles having mass \(m \ll M\) . Elastic collisions occur at random times as the continuous limit of a Poissonian sequence, i.e. with an exponentially decaying probability distribution of durations \(\tau \)  between collisions, \(P(\tau ) = \bar{\tau }^{-1}e^{-\tau /\bar{\tau }}\)  for the average collision rate \(1/\bar{\tau }\) . Collisions will alter the active particle's momentum instantaneously; between collisions the system evolves under (REF ) as in Section II above. The full evolution of the system is thus a piecewise deterministic process [1]} which can easily be simulated, and so we will begin our investigation of Hamiltonian active Brownian particles by exploring the qualitative effects of a Brownian environment on our Hamiltonian active particles within this collisional model. After identifying a potentially dramatic and counter-intuitive effect in which environmental dissipation can actually help the active particle travel farther than it otherwise would, we will simplify our model to pure viscous damping in order to obtain some more general results.
